Classification of Four-Dimensional CR Submanifolds of the Homogenous Nearly Kähler ${\mathbb{S}}^{\mathbf{3}}\mathbf{\times }{\mathbb{S}}^{\mathbf{3}}$ Which Almost Complex Distribution Is Almost Product Orthogonal on Itself

The product manifold ${\mathbb{S}}^3\times {\mathbb{S}}^3$, which belongs to the homogenous six-dimensional nearly Kähler manifolds, admits two structures, the almost complex structure J and the almost product structure P. The investigation of embeddings of different classes of CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ was started some time ago by investigating three-dimensional CR submanifolds. It resulted that the almost product structure P is very important for the study of CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$, since submanifolds characterized by different actions of the almost product structure on base vector fields often appear as a result of the study of some specific types of CR submanifolds. Therefore, the investigation of four-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ is initiated in this article. The main result is the classification of four-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$, whose almost complex distribution ${\mathcal{D}}_1$ is almost product orthogonal on itself. First, it was proved that such submanifolds have a non-integrable almost complex distribution, and then it was proved that these submanifolds are locally product manifolds of curves and three-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ of the same type, and they were therefore constructed in this way.